The value of the damping ratio ζ determines the behavior of the system. A damped harmonic oscillator can be:
(a) Overdamped (ζ > 1): The system returns (exponentially decays) to equilibrium without oscillating. Larger values of the damping ratio ζ return to equilibrium more slowly. (b) Critically damped (ζ = 1): The system returns to equilibrium as quickly as possible without oscillating. This is often desired for the damping of systems such as doors. (c) Underdamped (0 < ζ < 1): The system oscillates (at reduced frequency compared to the undamped case) with the amplitude gradually decreasing to zero. (d) Undamped (ζ = 0): The system oscillates at its natural resonant frequency (ωo).